C
1. Experiments with capacitors
1.1
Introduction
In many processes in physics, chemistry, and biology, the change in time of a quantity often is
proportional to its momentary value. Examples include:
• The number of decays of instable atoms in a radioactive sample within a certain time is
proportional to the number of instable atoms present in the sample.
• The change in concentration of a reactant during a chemical reaction is proportional to the
concentration.
• The growth rate of a given population is proportional to the number of individuals in that
population.
• A body moving through a viscous liquid experiences a force of friction (and thereby an
acceleration), which is proportional to its velocity.
• The current during the discharge of a capacitor across an electrical resistance is proportional
to the charge of the capacitor.
All these processes are quantitatively described using exponential functions. Using the example
of charging and discharging of a capacitor, we will introduce the bascis characteristics of exponential
functions. Some keywords:
• exponential function,
• electric circuits,
• Kirchhoff’s laws,
• charging and discharging of a capacitor, and
• measurement of electrical currents and voltages.
1
2
1. Experiments with capacitors
1.2
1.2.1
Theory
Charging and discharging of a capacitor
The generic electrical circuit of these experiments is shown in Fig. 1.1: The capacitor with capacity
C is charged across the (Ohmic) resistor R if the switch is in positions 1, and discharged when in
position 2.
R
1
V0
S
+
_
I
2
C
+
_
VC
Figure 1.1: Generic circuit.
Charging
Discharging
A power supply providing the DC voltage V0
is connected in series to a resistor R and a
capacitor C. At time t = 0, the switch S
is put into position 1; as a consequence, the
capacitor is charges via the resistor R. We
will derive an expression describing the time
dependence of voltage V (t) and current I(t).
The capacitor C is charged; the corresponding voltage between the two contacts be V0 .
At t = 0 the switch S is switched to position
2: the capacitor is discharged across the resistor R. Again, we are interested in voltage
V (t) and current I(t) as function of time.
The initial conditions at time t = 0 are:
Q(0) = 0
Q(0) = Q0 = V0 C
Using the first one of Kirchhoff’s laws we obtain in both cases:
V0 = I R +
Q
C
0 = IR +
Q
C
Inserting the relation I = dQ/dt yields:
V0 =
dQ
Q
R+
dt
C
(1.1)
0 =
dQ
Q
R+
dt
C
(1.2)
This means that the change in charge on the capacitor is proportional to the charge Q(t).
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1.2. THEORY
3
The solution of the inhomogeneous differential equation Eq. 1.1 is given by the sum of
the solution of the corresponding homogeneous equation Eq. 1.2 and one particular
solution of the inhomogeneous equation.
In the limit of very long times t → ∞ the
charge approaches the final value Q0 = V0 C:
The solution of the homogeneous equation
Eq. 1.2 is found by seperating the variables
Q and t:
dQ
dt
=−
.
Q
RC
Integration leads to:
ln Q = −
Q(t = ∞) = Q0 = V0 C.
t
+ ln A.
RC
Using the constant of integration ln A we obtain
t
Q(t) = A e− RC .
(1.3)
The general solution of Eq. 1.1 becomes:
t
Q(t) = A0 e− RC + Q0
Determination of the constants from the starting conditions:
Q(0) = A0 + Q0 = 0 ⇒ A0 = −Q0
t
− RC
Q(t) = Q0 (1 − e
Q(0) = Q0 = A
t
t
Q(t) = Q0 e− RC = V0 C e− RC
)
t
= V0 C (1 − e− RC )
Calculation of the current I = dQ/dt:
I(t) =
V0 − t
e RC
R
I(t) = −
(1.4)
I(t)
V0 − t
e RC
R
(1.5)
I(t)
V0
R
t
t
V0
R
Figure 1.2: Current during charging and discharging of a capacitor.
Calculation of the voltage VC = Q/C:
t
VC (t) = V0 (1 − e− RC )
(1.6)
t
VC (t) = V0 e− RC
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(1.7)
4
1. Experiments with capacitors
VC (t)
VC (t)
V0
V0
t
t
Figure 1.3: Voltage during charging and discharging of a capacitor..
Often, the characteristic timescale t0.9 (rise time) is given, which is defined as time needed to reach
90% of the value in the asymptotic limit VC = 0.9 V0 :
VC
V0
0.9 V0
t0.9
VC (t0.9 ) = 0.9 V0 = V0 (1 − e− RC )
t 0.9
Problem 1:
t
Figure 1.4: Definition of the rise time t0.9 .
• Determine t0.9 for given values of R and C.
• How should R and (or) C be changed in order to obtain a shorer rise time
t0.9 ?
1.2.2
Properties of the exponential function
The exponential function f (t) = A e−αt possesses the following properties:
• At any time t, the derivative df /dt with respect to time is proportional to the current value
f (t):
df
= −α A e−αt = −αf (t).
dt
The negative sign indicates that f (t) is monotonically decreasing.
• For equidistant steps ∆t, the value of f (t) always changes by the same factor at each step:
e−α ∆t .
The latter can easily be seen in a graphical representation of an exponential function, like in
Fig. 1.5.
The function f (t) = A e−αt reaches the value 0 in the asymptotic limit t → ∞. For this reason,
processes described by a decreasing exponential function are characterized by a time called half-life
T1/2 or alternatively by the characteristic time constant τ .
Laboratory Manuals for Students in Biology and Chemistry - Course PHY127
1.2. THEORY
5
f(t)
A
A
9
A
3
A
27
Δt
2Δt
3Δt
t
Figure 1.5: Exponential function: decrease by a factor of one third in each interval δt.
Half-life:
After the half-life period T1/2 the function f (t) is dropped by half of its initial value:
f(t)
f (0) = A
A
f (T1/2 ) =
= A e−αT1/2
2
1
ln
= −α T1/2
2
ln 2
T1/2 =
α
A
A
e
α1<α2
α2
(T1/2)2 (T1/2)1
Figure 1.6: Illustration of the half-life.
t
(Characteristic) Time constant:
After a time corresponding to the time constant τ the function f (t) reaches 1/e of its initial value:
f (0) = A
A
f (τ ) =
= A e−1 = A e−ατ
e
1
τα = 1
⇒
τ =
α
Figure 1.7: Illustration of the time constant.
f(t)
A
A
e
e = 2.7183...
1/e = 0.3679...
α2
α1<α2
τ2
τ1
Laboratory Manuals for Students in Biology and Chemistry - Course PHY127
t
6
1. Experiments with capacitors
Ex.: Discharging of a capacitor
According to Eq. 1.6) the voltage behaves as:
t
= V0 e− RC
1
α =
RC
VC
ergo:
τ
= RC
T1/2 = RC ln 2
Figure 1.8: Discharging of a capacitor.
VC
V0
(1.8)
(1.9)
V0
2
V0
e
T1/2 τ
Laboratory Manuals for Students in Biology and Chemistry - Course PHY127
t
1.3. EXPERIMENTAL PART
1.3
1.3.1
7
Experimental Part
Goal of this experiment
A capacitor is charged and discharged through a resistor R. Voltage across the capacitor and the
current through the resistor are monitored as function of time for two different RC time constants
τ1 ' 10s and τ2 ' 15s.
1.3.2
Setup
R
1
S
V0
I
2
C1
C2
C3
Ci
VC
Figure 1.9: Electric circuit.
The power supply and volt- and amperemeter are connected to the corresponding jacks. Using the
switches On-Off (Ein-Aus) the number of capacitors can be varied. Since the capacitors are connected in parallel, the total capacitance of the circuit is given by the sum of the single capacitances:
Ctot =
P
Ci
The resistor R is to be plugged onto the corresponding jacks. Choose a combination of R and C
which leads to a time constant RC ' 10..15 s according to Eq. 1.8.
Internal resistance of the voltmeter
Every real voltmeter has a finite internal resistance RV . If we take this internal resistance into
account we obtain the following circuit:
R
I
V0
C
VC
RV
The voltmeter used here has an internal
resistance of 107 Ω. Therefore RV R,
i.e. the current passing through the
voltmeter can be neglected here.
Figure 1.10: Internal resistance of the
voltmeter.
Laboratory Manuals for Students in Biology and Chemistry - Course PHY127
8
1. Experiments with capacitors
1.3.3
Assembly
• Connect the power supply and the instruments.
• Choose the time constant and the corresponding combination of R and C.
• Plug in the resistor R.
• Activate the capacitors required.
• Turn on the voltage and set the voltage to 15 V.
1.3.4
Measurements
Charging the capacitor (Switch in position 1):
• The voltage across the capacitors is to be measured as function of time after switching to
position 1 (t = 0): for doing so read and note the voltage VC every 5 s.
• Discharge the capacitors.
• For the second time, record the current as function of time in the same way than the voltage
before. In particular, pay attention to the current at t = 0 when switching on.
Problem 2: Why the voltage does not reach V0 ?
Discharging the capacitor (Switch in position 2):
• Charge the capacitors up to a voltage V .
• Put the switch in position 2 and record voltage and current as function of time in the same
way as during charging.
1.3.5
Report
• Answer the questions in the text.
• Briefly resume the goal and the experiment itself. In particular, mention the following topics:
– Characteristic properties of the exponential function.
– Meaning of half-life and time constant τ
• Analysis:
– Plot the voltage VC (t) and the current I(t) on linear graph paper.
Laboratory Manuals for Students in Biology and Chemistry - Course PHY127
1.3. EXPERIMENTAL PART
9
– Moreover, plot the decreasing curves (current for charging, voltage for discharging) on
semi-logarithmic paper.
– Determine the time constants τ from these plots as outlined below. Compare the values
obtained with the values calculated from the resistance and the total capacitance.
Hint: If you use semi-logarithmic graph paper, the slopes are given by the difference of the
logarithms of the values plotted on the logarithmic axis, as shown in Fig. 1.12. Alternatively,
the values ln V (t) and ln I(t) may be plotted against a linear scale (normal linear graph paper)
if no semi-logarithmic paper is at disposal.
Determination of the time constant from the plots
• From the linear representation of the exponantial function:
I
As shown in the figure, the characteristic time
constant can be read directly from the plot.
V0
R
V0
R·e
τ
Figure 1.11: Determination of the time constant
τ from the linear plot.
t
• From the logarithmic representation of the exponential function:
Logarithmizing
I =
leads to
ln
V0 − t
e RC
R
RI
t
=−
.
V0
RC
Note: Only values can be logarithmized, not units. Therefore, the units must cancel. Then
we can rewrite the equation:
t
R
ln I + ln
= −
.
(1.10)
V0
RC
Using the logarithm at base 10, we obtain the same relation except for a correction factor
log e:
IR
t
log
= −
log e
V0
RC
and, after the units having canceled:
log I + log
R
t
= −
log e.
V0
RC
Laboratory Manuals for Students in Biology and Chemistry - Course PHY127
(1.11)
10
1. Experiments with capacitors
Equations (1.10) and (1.11) represent straight lines. For the slopes we obtain:
∆(ln I)
∆t
= −
∆(log I)
∆t
1
RC
log e
RC
∆t RC = log e ∆(log I) ∆t RC = ∆(ln I) ln I
ln
= −
log I
log
V0
R
Δ (ln I)
V0
R
Δ (log I)
Δt
Δt
t
t
Figure 1.12: Calculations of the slopes using natural (left-hand-side) and common logarithm (righthand-side).
Demonstration
The assistant will demonstrate sawtooth oscillations (also termed relaxation oscillations) using
several different combinations of resistor and capacitor in the electric circuit shown in Fig. 1.3.5.
Observe the changes in frequency and ignition time. The circuit is explained in detail in the
appendix.
The voltage VC (t) is measured on the oscilloscope.
R
V0
C
GL
KO
Figure 1.13: Electric circuit used for the
demonstration. The polarity of the bulb is important!
Laboratory Manuals for Students in Biology and Chemistry - Course PHY127
1.4. APPENDIX
1.4
11
Appendix
1.4.1
Generation of sawtooth oscillations using a glow lamp
A glow lamp is a bulb filled with gas and equipped with two electrodes.
I
disk and ring
electrodes
two wire
electrodes
VL
VZ
V
Figure 1.15: I − V -curve of a glow lamp. VZ
denotes the voltage at which the lamp ignites, VL the voltage at which the lamp goes
off.
Figure 1.14: Types of glow lamps.
A voltage is applied to the electrodes. Free electrons present in the gas are accelerated by the electric
field. If the field is sufficiently strong, the electrons acquire enough energy between two collisions to
ionize gas atoms. The number of free charges, thereby, increases generating an avalanche of charge
carriers and ionized atoms. The current is large and the inner resistance Ri of the bulb decreases
strongly. The critical field or voltage is called ignition voltage of the bulb VZ .
Circuit for generating sawtooth oscillations
R
V0
S
C
GL
Figure 1.16: Electric circuit.
KO
A capacitor C is connected in parallel to
a glow lamp, both are linked to a voltage
source through a resistor R. The resistor
R shall be large compared to the inner resistance of the ignited bulb Ri . As long as
the current through the bulb is small, i.e.
VC < VZ , the capacitor is slowly charged
through the resistor R. When the voltage
VC reaches the ignition voltage, the bulb ignites and the capacitor is rapidly discharged
through the bulb Ri .
Since R Ri , the re-charging of the capacitor through R is slow and VC decreases until it reaches
the voltage VL , at which the bulb goes off. Then, the capacitor start charging again and the whole
cycle repeats. If the voltage VC is measured using an oscilloscope we will see the following curve:
(compare to Fig. 1.3):
Laboratory Manuals for Students in Biology and Chemistry - Course PHY127
12
1. Experiments with capacitors
VC
V0
TB = bulb is ignited,
T2 − T1 = recharging of the capacitor from
VL to VZ .
VZ
VL
TB
T1
Figure 1.17: Voltage of the capacitor.
t
T2
Calculation of the frequency
Charging of the capacitor according to Eq. 1.6)):
V0 − VL
V0
T2
V
− VZ
0
= V0 (1 − e− RC ) ⇒
V0
T1
T1
VL = V0 (1 − e− RC ) ⇒
= e− RC
VZ
= e− RC
t2
Division of both equations and logarithmizing leads to
1
V0 − VL
= e− RC ·(T1 −T2 )
V0 − VZ
ln
V0 − VL
V0 − VZ
T2 − T1
1
· (T1 − T2 )
RC
V0 − VL
= RC ln
.
V0 − VZ
= −
In the case R Ri , we may neglect the ignition time TB with respect to T2 − T1 . Hence we obtain
for the frequency ν of the oscillations:
ν ≈
1
1
1
1
=
≈
V
−V
0
L
T1 − T2
RC ln V −V
RC
0
Z
Laboratory Manuals for Students in Biology and Chemistry - Course PHY127
1.4. APPENDIX
1.4.2
13
Some important applications of the exponential function
a) Radioactive decay
N (t)
N0
N (t) = N0 e-α t
N0 = number of unstable nuclei at time t = 0
N (t) = number of unstable nuclei at time t
At half-time T1/2 , half of the nuclei did not yet
decay.
N0
2
T1/2
Figure 1.18: Radioactive decay.
t
Depending on the stability of the isotopes, typical half-life times range from 10−22 seconds up to
thousands of years.
b) Absorption of light in matter (electromagnetic radiation including γ-rays)
I (x)
I0
I (x) = I0 e-μ x
light
I(x)
I0
I0
2
x=0
d1/2
x
x
x
Figure 1.19: Absorption of electromagnetic
radiation.
Be I0 the intensity at x = 0. I(x) denotes the intensity of the radiation after having passed a layer
of thickness x of the material. d1/2 corresponds to the thickness of the layer at which 50% of the
radiation is absorbed.
Note: The intensity of γ−-radiation never reaches zero even for very thick radiation shielding.
c) Motion in viscous liquids
In viscous liquids the friction is proportional to and opposite to the speed. The equation of motion
of a body moving through a liquid reads:
m
d2 x
dvx
= m
= −β vx + F0
2
dt
dt
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14
1. Experiments with capacitors
v(t)
F0 = driving (constant) force, e.g. gravity or
electric field
βvx = friction in a viscous liquid
v0 = stationary speed
v0
v(t) = v0 (1- e- /m t)
b
t
Figure 1.20: Motion through a viscous liquid.
d) In biology
In many biological processes like decomposition, resorption, or excretion, for instance, several
processes following exponential functions with different time constants are superimposed in complex
ways.
Laboratory Manuals for Students in Biology and Chemistry - Course PHY127